### Polynomial Roots and Calabi-Yau Geometries

"... The examination of roots of constrained polynomials dates back at least to Waring and to Littlewood. However, such delicate structures as fractals and holes have only recently been found. We study the space of roots to certain integer polynomials arising naturally in the context of Calabi-Yau spaces ..."

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The examination of roots of constrained polynomials dates back at least to Waring and to Littlewood. However, such delicate structures as fractals and holes have only recently been found. We study the space of roots to certain integer polynomials arising naturally in the context of Calabi-Yau spaces, no-tably Poincare ́ and Newton polynomials, and observe various salient features and geometrical patterns.

### Classification and Properties of Hyperconifold Singularities and Transitions

"... This paper is a detailed study of a class of isolated Gorenstein threefold singularities, called hyperconifolds, that are finite quotients of the conifold. First, it is shown that hyperconifold singularities arise naturally in limits of smooth, compact Calabi–Yau threefolds (in particular), when the ..."

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This paper is a detailed study of a class of isolated Gorenstein threefold singularities, called hyperconifolds, that are finite quotients of the conifold. First, it is shown that hyperconifold singularities arise naturally in limits of smooth, compact Calabi–Yau threefolds (in particular), when the group action on the covering space develops a fixed point. The Zn-hyperconifolds—those for which the quotient group is cyclic—are classified, demonstrating a one-to-one correspondence between these singularities and three-dimensional lens spaces L(n, k), which occur as the vanishing cycles. The classification is constructive, and leads to a simple proof that a Zn-hyperconifold is mirror to an n-nodal variety. It is then argued that all factorial Zn-hyperconifolds have crepant, projective resolutions, and this gives rise to transitions between smooth compact Calabi–Yau threefolds, which are mirror to certain conifold transitions. Formulae are derived for the change in both fundamental group and Hodge numbers under such hyperconifold transitions. Finally, a number of explicit examples are given, to illustrate how to construct new Calabi–Yau manifolds using hyperconifold transitions, and also to highlight the differences which can occur when these singularities occur in non-factorial varieties.

### Hypercharge Flux in Heterotic Compactifications

"... We study heterotic Calabi-Yau models with hypercharge flux breaking, where the visible E8 gauge group is directly broken to the standard model group by a non-flat gauge bundle, rather than by a two-step process involving an intermediate GUT theory and a Wilson line. It is shown that the required alt ..."

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We study heterotic Calabi-Yau models with hypercharge flux breaking, where the visible E8 gauge group is directly broken to the standard model group by a non-flat gauge bundle, rather than by a two-step process involving an intermediate GUT theory and a Wilson line. It is shown that the required alternative E8 embeddings of hypercharge, normalized as required for gauge unification, can be found and we classify these possibilities. However, for all but one of these embeddings we prove a general no-go theorem which asserts that no suitable geometry and vector bundle leading to a standard model spectrum can be found. Intuitively, this happens due to the large number of index conditions which have to be imposed in order to obtain a correct physical spectrum in the absence of an underlying GUT theory.

### A Closer Look at Mirrors and Quotients of

, 2013

"... Let X be the toric variety (P1)4 associated with its four-dimensional polytope ∆. Denote by X ̃ the resolution of the singular Fano Xo variety associated with the dual polytope ∆o. Generically, anticanonical sections Y of X and anticanonical sections Y ̃ of X ̃ are mirror partners in the sense of Ba ..."

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Let X be the toric variety (P1)4 associated with its four-dimensional polytope ∆. Denote by X ̃ the resolution of the singular Fano Xo variety associated with the dual polytope ∆o. Generically, anticanonical sections Y of X and anticanonical sections Y ̃ of X ̃ are mirror partners in the sense of Batyrev. Our main result is the following: the Hodge-theoretic mirror of the quotient Z associated to a maximal admissible pair (Y,G) in X is not a quotient Z ̃ associated to an admissible pair in X̃. Nevertheless, it is possible to construct a mirror orbifold for Z by means of a quotient of a suitable Y ̃. Its crepant resolution is a Calabi-Yau threeefold with Hodge numbers (8, 4). Instead, if we start from a non-maximal admissible pair, in same cases, its mirror is the quotient associated to an admissible pair. 1

### Contents

, 1998

"... Abstract. These are the notes of two series of talks about Givental’s proof of the mirror conjecture for projective complete intersections, given by the authors at the Università di Roma “La Sapienza”, and at the Scuola Normale ..."

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Abstract. These are the notes of two series of talks about Givental’s proof of the mirror conjecture for projective complete intersections, given by the authors at the Università di Roma “La Sapienza”, and at the Scuola Normale